Answer :
Answer:-
To simplify the expression \( \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} \), let's break it down step by step:
1. **Simplify each square root individually:**
- \( \sqrt{10} \) cannot be simplified further since 10 is not a perfect square.
- \( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \).
- \( \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10} \).
- \( \sqrt{5} \) cannot be simplified further.
- \( \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} \).
2. **Substitute the simplified forms into the original expression:**
\[ \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} = \sqrt{10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5} \]
3. **Combine like terms:**
- Combine \( \sqrt{10} \) and \( 2\sqrt{10} \): \( \sqrt{10} + 2\sqrt{10} = 3\sqrt{10} \).
- Combine \( -\sqrt{5} \) and \( -4\sqrt{5} \): \( -\sqrt{5} - 4\sqrt{5} = -5\sqrt{5} \).
- The \( 2\sqrt{5} \) term stands alone.
So the expression simplifies to:
\[ 3\sqrt{10} + 2\sqrt{5} - 5\sqrt{5} \]
4. **Further simplify the terms involving \( \sqrt{5} \):**
\[ 2\sqrt{5} - 5\sqrt{5} = -3\sqrt{5} \]
Therefore, the simplified expression is \( 3\sqrt{10} - 3\sqrt{5} \).